Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











FresnelC






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > FresnelC[z] > Series representations > Generalized power series > Expansions at generic point z==z0 > For the function itself





http://functions.wolfram.com/06.33.06.0014.01









  


  










Input Form





FresnelC[z] == Pi^(3/2) Sum[((2^(2 k - 3/2) Subscript[z, 0]^(1 - k))/k!) HypergeometricPFQRegularized[{1/4, 3/4, 1}, {(2 - k)/4, (3 - k)/4, 1 - k/4, (5 - k)/4}, -((Pi^2 Subscript[z, 0]^4)/16)] (z - Subscript[z, 0])^k, {k, 0, Infinity}]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["FresnelC", "[", "z", "]"]], "\[Equal]", RowBox[List[SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["2", " ", "k"]], "-", FractionBox["3", "2"]]]], " ", SubsuperscriptBox["z", "0", RowBox[List["1", "-", "k"]]]]], RowBox[List["k", "!"]]], RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "4"], ",", FractionBox["3", "4"], ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["2", "-", "k"]], "4"], ",", FractionBox[RowBox[List["3", "-", "k"]], "4"], ",", RowBox[List["1", "-", FractionBox["k", "4"]]], ",", FractionBox[RowBox[List["5", "-", "k"]], "4"]]], "}"]], ",", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["\[Pi]", "2"], " ", SubsuperscriptBox["z", "0", "4"]]], "16"]]]]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], "k"]]]]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <semantics> <mi> C </mi> <annotation encoding='Mathematica'> TagBox[&quot;C&quot;, FresnelC] </annotation> </semantics> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <msup> <mi> &#960; </mi> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <mrow> <msup> <mn> 2 </mn> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> - </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <msubsup> <mi> z </mi> <mn> 0 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> k </mi> </mrow> </msubsup> </mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 3 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 4 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> - </mo> <mi> k </mi> </mrow> <mn> 4 </mn> </mfrac> <mo> , </mo> <mfrac> <mrow> <mn> 3 </mn> <mo> - </mo> <mi> k </mi> </mrow> <mn> 4 </mn> </mfrac> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mi> k </mi> <mn> 4 </mn> </mfrac> </mrow> <mo> , </mo> <mfrac> <mrow> <mn> 5 </mn> <mo> - </mo> <mi> k </mi> </mrow> <mn> 4 </mn> </mfrac> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msubsup> <mi> z </mi> <mn> 0 </mn> <mn> 4 </mn> </msubsup> </mrow> <mn> 16 </mn> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;3&quot;, TraditionalForm]], SubscriptBox[OverscriptBox[&quot;F&quot;, &quot;~&quot;], FormBox[&quot;4&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[&quot;1&quot;, &quot;4&quot;], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;3&quot;, &quot;4&quot;], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;1&quot;, HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List[&quot;2&quot;, &quot;-&quot;, &quot;k&quot;]], &quot;4&quot;], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[RowBox[List[&quot;3&quot;, &quot;-&quot;, &quot;k&quot;]], &quot;4&quot;], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, FractionBox[&quot;k&quot;, &quot;4&quot;]]], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[RowBox[List[&quot;5&quot;, &quot;-&quot;, &quot;k&quot;]], &quot;4&quot;], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], &quot;;&quot;, TagBox[RowBox[List[&quot;-&quot;, FractionBox[RowBox[List[SuperscriptBox[&quot;\[Pi]&quot;, &quot;2&quot;], &quot; &quot;, SubsuperscriptBox[&quot;z&quot;, &quot;0&quot;, &quot;4&quot;]]], &quot;16&quot;]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] </annotation> </semantics> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> FresnelC </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <power /> <pi /> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> <apply> <power /> <apply> <factorial /> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> HypergeometricPFQRegularized </ci> <list> <cn type='rational'> 1 <sep /> 4 </cn> <cn type='rational'> 3 <sep /> 4 </cn> <cn type='integer'> 1 </cn> </list> <list> <apply> <times /> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> 3 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> k </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> 5 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </list> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <cn type='integer'> 16 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["FresnelC", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["2", " ", "k"]], "-", FractionBox["3", "2"]]]], " ", SubsuperscriptBox["zz", "0", RowBox[List["1", "-", "k"]]]]], ")"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "4"], ",", FractionBox["3", "4"], ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["2", "-", "k"]], "4"], ",", FractionBox[RowBox[List["3", "-", "k"]], "4"], ",", RowBox[List["1", "-", FractionBox["k", "4"]]], ",", FractionBox[RowBox[List["5", "-", "k"]], "4"]]], "}"]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "16"]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[Pi]", "2"], " ", SubsuperscriptBox["zz", "0", "4"]]], ")"]]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "k"]]], RowBox[List["k", "!"]]]]]]]]]]]










Date Added to functions.wolfram.com (modification date)





2007-05-02